Abstract. We consider the joint distributions of particle positions for the
continuous time totally asymmetric simple exclusion process
(TASEP). They are expressed as Fredholm determinants with a kernel
defining a signed determinantal point measure. We then consider
certain periodic initial conditions and determine the kernel in
the scaling limit. This result has been announced first in a
letter by one of us and here we provide a
self-contained derivation. Connections to last passage directed
percolation and random matrices are also briefly discussed.